AbstractAims. We aim to understand the interplay between
non-radial oscillations and stellar magnetic activity and test the
feasibility of doing asteroseismology of magnetically active stars. We
investigate the active slow rotator EK Eri which is the likely
descendant of an Ap star. Methods. We analyze 30 years of photometric
time-series data, 3 years of HARPS radial velocity monitoring,
and 3 nights of high-cadence HARPS asteroseismic data. We construct a
high-S/N HARPS spectrum that we
use to determine atmospheric parameters and chemical composition.
Spectra observed at different rotation phases are analyzed to search
for signs of temperature or abundance variations. An upper limit on the
projected rotational velocity is derived from very high-resolution CES
spectra. Results. We detect oscillations in EK Eri
with a frequency of the maximum power of Hz, and we
derive a peak amplitude per radial mode of 0.15 m s-1,
which is a factor of 3
lower than expected. We suggest that the magnetic field may act to
suppress low-degree modes. Individual frequencies can not be extracted
from the available data. We derive accurate atmospheric parameters,
refining our previous analysis, finding K,
,
and metallicity .
Mass and radius estimates from the seismic analysis are not accurate
enough to constrain the position in the HR diagram and the evolutionary
state. We confirm that the main light variation is due to cool spots,
but that other contributions may need to be taken into account. We
tentatively suggest that the rotation period is twice the photometric
period, i.e., d,
and that the star is a dipole-dominated oblique rotator viewed close to
equator-on. We conclude from our derived parameters that km s-1
and we show that the value is too low to be reliably measured. We also
link the time series of direct magnetic field measurements available in
the literature to our newly derived photometric ephemeris.

1 Introduction

EK Eri (HR 1362, HD 27536) is a unique case of a
slowly rotating (
km s-1)
G8 giant or sub-giant, which is over-active with respect to its
rotation rate and evolutionary state. It is exhibiting brightness
variations with a period of more than 300 days, believed to be due to
semi-stable star spots being rotated across the projected surface (Strassmeier et al. 1999).
It has been suggested that the associated strong magnetic field is a
fossil remnant from
its main sequence life as a magnetic Ap star (Stepien
1993).

The star has been monitored photometrically since 1978, with
first results included in Strassmeier
et al. (1990).
In a subsequent study, Strassmeier
et al. (1999) derived a photometric period of
306.9 d from twenty years of monitoring, but noted that the
light curve could
be split into two segments with different periods, 311 d
(pre-1987) and 294 d (post-1992) respectively, and with a
period of relatively small light variations in between,
possibly reflecting two distinct magnetic cycles. These photometric
data also showed that the star gets redder when it gets fainter, which
agrees with cool spots as the cause of the light variation. From
high-resolution spectroscopy, Strassmeier
et al. (1999) also determined
the fundamental stellar parameters and showed from the radius-
constraints that the star must be
seen close to equator-on, i.e. .

Aurière et al.
(2008) published the first direct measurement of the magnetic
field of EK Eri. They find the field
to be large scale, rather than a solar-type (small-scale) field,
dominated by a poloidal mostly axisymmetric component, resembling a
dipole with a strength
of 270 G.
They also observe some modulation over the rotation period, although
their data do not cover the full
photometric period. Their results strengthen the interpretation that
EK Eri is a descendant of a slowly rotating magnetic
Ap star which is now approaching the giant branch.

In Dall et al.
(2005, hereafter Paper I) we refined the fundamental
parameters of EK Eri using new HARPS spectra and found
radial velocity (RV) variations with peak-to-peak amplitude of 100 m s-1.
This variation was shown to correlate extremely well with the calcium H
& K activity index (
)
as well as with the bisector inverse
slope (BIS) of the cross-correlation function. However we found a
positive correlation for the BIS rather than
the negative correlation expected from spot-induced RV variations
(e.g., Desort et al. 2007).
Such correlations have previously
been attributed to fainter stellar companions contributing to the
signal, effectively disguising the signal
of a planetary companion (Santos et al. 2002;
Zucker
et al. 2004).

In this paper we present the updated results from 30 years of
photometric monitoring, as well as from three years of RV monitoring,
and three half-nights of high-cadence RV measurements used for an
asteroseismic analysis.

In Sect. 2
we present our new observations and the data reduction. In
Sect. 3
we present our results, which we discuss in Sect. 4.
In Sect. 5
we give a summary of our conclusions.

2 Observations

2.1 Photometry

We present new photometric data for the years 1998 through 2009.
Photometry from 1978 through early 1998 was previously analyzed by
Strassmeier et al.
(1999) and consisted of data from
various sources listed therein. The new data comes from Amadeus, one
of the two 0.75m Vienna ``Wolfgang-Amadeus'' twin automatic
photoelectric telescopes located at Fairborn Observatory in
Washington Camp in southern Arizona (Strassmeier
et al. 1997).
Since late 1998, 1719 nightly V and I
data points were acquired, each
the mean of three measurements of EK Eri and a comparison
star. These data were transformed to the Johnson-Cousins system and
used EK Eri as the comparison star and
EK Eri as the check star. The data quality varied
between
an external rms of 4 mmag to 7 mmag in V. Details
of the data
reduction procedure were previously given by Granzer
et al. (2001).
The main part of the data set is shown in Fig. 1.

Figure 1:

The photometric time series in V from 1979-2009 and
in V-I from 1996-2009. The
expected times of V maxima are indicated with
dashed lines. The inset shows the power spectrum from the period
analysis (see Sect. 3.1).

The Wolfgang telescope was used on five dedicated nights 20-24
December 2007 to monitor the star with a cadence of 3.0 min to
check for possible high-frequency photometric variations. The
second night was lost due to poor weather, the third night was usable
for about half of the night, the others were good. On the three nights
20th, 23rd and 24th, the target was observed for 7.5 h
continuously, on the 21st for 4.5 h, and on the 22nd for
2 h. The total time on target during these five nights was
29 h of which the 2 h on the 22nd were of poor
quality and discarded.
A total of 542 y data points were acquired.
Only Strömgren y was employed.
These data were calibrated to the standard Strömgren magnitudes.
The external rms of the time-series is 1.2 mmag in y.

2.2 Spectroscopy

We have been conducting ultra-stable radial velocity monitoring of
EK Eri with HARPS (Mayor et al. 2003; Rupprecht
et al. 2004) from September 2004 to April 2007,
mostly with rather uneven spacing between data points.
Typical exposure times were 480-600 s resulting in a typical
peak S/N 300
for individual exposures, which translates to
an intrinsic RV precision of better than 1 m s-1
per exposure. The HARPS resolution is R
= 100 000.

Accurate radial velocities are derived by the HARPS pipeline
using cross-correlation with line templates (masks).
The high-precision comes from the use of a simultaneous Th-Ar
calibration spectrum for accurate wavelength calibration and from the
intrinsic high stability of the spectrograph. The resulting
cross-correlation
function (CCF) is fitted with a Gaussian to obtain the RV.

In March 2007 we had three consecutive half-nights of
continuous monitoring
to search for oscillations and probe the asteroseismic parameters of
the star. We used exposure times of 360-480 s which resulted
in peak
S/N 250-300
for individual exposures at a cadence of about 8 min. We
observed for about three hours per night,
obtaining a total of 70 spectra.

We have also retrieved archive data of EK Eri taken
with the CES high-resolution spectrograph at ESO/La Silla
Observatory. One 900 s spectrum at a central wavelength of
6155 Å was obtained on UT date 2004-09-03 and one
600 s spectrum at 6463 Å was obtained on UT date
2005-08-11. The spectra cover 39 Å and 43 Å,
respectively.
These data were reduced using the CES pipeline.
The resolution is
with
in both spectra.

3 Results

In the following we will investigate the high-resolution spectroscopy
and long-term photometric time-series data. The numerical results are
summarized in Table 1.

3.1 Photometric period and modulation

Strassmeier et al.
(1999) determined the period of the long-term photometric
variation of EK Eri based on the data obtained to that date.
They also found that
the period may have changed slightly when comparing the years 1978-1986
to the years 1992-1999, possibly due to distinct activity cycles. Here
we derive the period using the full data set,
using a variety of techniques summarized in Table 2. From these
results we adopt a photometric period of d
(see inset of Fig. 1).
Given the errors on the individual results and the obvious spot
evolution, we
cannot comfortably give the period with higher precision.
The corresponding ephemeris is given in Table 1 and is marked
by dashed vertical lines in Fig. 1. As can be
seen, the ephemeris can not always reproduce the data, indicating that
the spots are changing size and
configuration. From Fig. 1
it is clear that when the star is fainter, V-I
is higher (i.e., the star is redder)
which points to cool spots as the cause of the variation. Since the
period is likely much longer than typical spot lifetimes we cannot hope
to extract any quantitative
information on the amount of differential rotation.

Table 2:
Results from different methods of period analysis of the long-term
photometry.

We will now describe the methods used to derive the periods in
Table 2.
Both the Discrete Fourier Transform (DFT; e.g., Reegen 2007) and Period04 (Lenz & Breger 2005) were
used, where we prewhitened
with two long periods corresponding to long-term drifts. Period04 is
developed for asteroseismic applications where the light curve can be
assumed to be made up of sinusoidal components. This makes it
inappropriate for our analysis of the long term variations which are
changing in amplitude and possibly also in period. The usual
formula for the period precision based on a mix of white and 1/f
noise therefore has no physical meaning.
The CLEAN algorithm (also known as iterative sine-wave fitting; e.g., Frandsen et al. 1995)
revealed only one main peak in the resulting amplitude spectrum after
pre-whitening with the long-term drifts.
The Lomb-Scargle periodogram gives results similar to the DFT, but with
slightly lower error.
The Lafler-Kinman analysis (Lafler
& Kinman 1965) gives a similar result, but the
periodogram is not as convincing - a trait shared by the minimum string
length (MSL) method. The phase dispersion minimization (PDM)
tend to generate higher period aliases, but even so the peak was
clearly identified, although not very well
determined.

3.2 Asteroseismic results

Figure 2:

RV vs. BJD for the three nights of high-cadence spectra used for the
asteroseismic analysis. The mean error bar is shown at the
lower left corner.
The three-night run displays clear variation, but with period
variations from
night-to-night, likely due to mode beating.

From its position in the HR diagram, EK Eri is expected to
show p-mode
oscillations that are stochastically excited and damped by near surface
convection similar to what is observed in the Sun and other solar-like
stars (Bedding &
Kjeldsen 2003).
We therefore carried out high-cadence (
min) spectroscopic
monitoring during three consecutive nights using HARPS as outlined in
Sect. 2.2.
In addition, five nights of high-cadence photometric monitoring was
conducted as described in Sect. 2.1

3.2.1 Radial velocity frequency spectrum

The time series is shown in Fig. 2 and shows
significant variability of a few metres per second
(peak-to-peak) with periods of roughly 1 h. The slight
night-to-night offsets are likely due to slow variations in the overall
activity level.
In Fig. 3
we show the Fourier power spectrum of the time series,
which shows the variability as an excess power in the frequency range
200-400Hz.
This excess agrees with the expected frequency of maximum power
estimated from scaling the solar value
(Kjeldsen &
Bedding 1995),
which gives roughly 340 Hz. The power at very low frequency (Hz) is
predominantly caused by the slow linear trend seen on the third night
and is probably not due to oscillations.

Figure 3:

Fourier spectrum of the three nights of HARPS high-cadence radial
velocity data.
The inset shows the spectral window with the same frequency scale as
the main panel.

The frequency spectra of the Sun and other solar-like stars show an
almost regular series of peaks. From this, one can extract the spacing
between modes of successive radial order called the large separation,
,
which provides a very precise measure of the mean density of
the star. By scaling the solar value we find that the expected large
separation
of EK Eri (using the scaling formula from Kjeldsen & Bedding 1995)
is around 20 Hz.

Due to the short and sparse coverage the present data set does
not allow
detection of the individual frequencies or the large separation.
However, we are able to estimate the amplitude per mode from the
excess power in the Fourier spectrum using the approach by
Kjeldsen et al. (2005).
First, the Fourier spectrum is converted into power
density by dividing by the area under the spectral window. Then we
convolve the
spectrum with a Gaussian with a width of ,
to create just a
single smooth hump of excess power and finally we multiply by
(Kjeldsen et al. 2008)
and take the square root to get the
amplitude per radial mode.

Figure 4:

Amplitude per radial mode. The amplitude is significantly lower than
expected - compare e.g. with Fig. 8 of Kjeldsen et al. (2008).
See text for explanation.

The result is shown in Fig. 4. The peak
amplitude per
radial mode is approximately 0.15 m s-1,
but with a large variation from
night-to-night, which is expected from the stochastic nature of the
excitation mechanism. Still, this amplitude is significantly lower than
expected (by at least a factor of 3) based on results
for stars with similar oscillation periods (Fig. 8 of Kjeldsen et al. 2008).
Part of this discrepancy
may be explained by mode beating in conjunction with the short time
span of our observations.
From this analysis we find the position
of maximum power to be Hz, where we
have adopted a 10% uncertainty using simulated data: for this purpose
we use the simulator described in
Stello et al. (2004).
The input mode frequencies were taken from a pulsation
model of the star derived using the ADIPLS code (Christensen-Dalsgaard
2008). We
calculated the individual mode amplitudes by scaling the shape of the
solar
excess
power by the acoustic cut-off frequency and normalised to the peak
amplitude found from Fig. 4. The input
amplitude takes
into account the different inertia of the modes and visibilities of
whole-disk integrated observations. The simulations show a 10% scatter
in
and a
30% scatter
in amplitude with assumed mode lifetimes in the range
1-20 days.
In addition, we tried to utilise the simulations to obtain a mode
lifetime,
but concluded that the data did not allow a robust estimate.

Figure 5:

A section of the combined HARPS spectrum illustrating the high S/N.
The observed and synthetic spectra are shown with solid and dot-dashed
lines, respectively.
The main lines contributing to the spectrum are marked by vertical
lines. The solid vertical lines mark the least blended lines that are
used in the abundance analysis.

We also searched for variability due to oscillations in the
high-cadence photometric data.
Of the three full nights obtained, a period analysis reveals no
periodicity on two of the nights, while on a third night a single peak
with a false alarm probability of 9.4% was detected at f=26.016 d-1
(301.11 Hz)
or around 55 min at an amplitude of 5 mmag. This
frequency corresponds well to the envelope found from the spectroscopic
time series.
However, based on the scaling relations of Kjeldsen & Bedding
(1995) and the observed RV amplitude, the photometric
amplitude of EK Eri is expected to be 50 ppm, which is a
factor of 100 lower than
the observed peak. Although this peak appears significant, we believe
it to be an artifact.

3.3 Effective temperature, gravity, abundances

To determine the fundamental parameters of EK Eri we analysed
a co-added spectrum with high S/N.
This spectrum was made from the three consecutive nights of
asteroseismic observations.
We shifted the HARPS spectra by the individually measured velocity
shifts and calculated the sum.
The blue arm of the spectrum from 4402-5304 Å has S/N
of 700 and the red arm from 5337-6905 Å has
S/N of 1300 in the continuum. We
made a careful normalization
by identifying continuum points in a synthetic
spectrum with the same parameters as EK Eri.
A small part of the co-added spectrum is shown in Fig. 5. Also plotted
is a computed synthetic spectrum which shows some discrepancies,
likely due to missing lines (e.g. 5008.7 Å) or erroneous
values (e.g. 5002.6 Å).

We used VWA to automatically select 1019 of the least blended
lines.
However, as part of the analysis we corrected the oscillator strengths ()
by measuring abundances for the same lines found in a spectrum of the
Sun from the Atlas by Hinkle
et al. (2000) (originally from Kurucz et al. 1984):
627 lines were found in common with the Sun and could be corrected,
and only these lines were considered in the further analysis.
This differential analysis
leads to a significant improvement in the rms scatter of the abundances
of Fe I and Fe II
from 0.12 to 0.06 dex. For other elements the improvement in the
scatter is typically 30%.

The fundamental atmospheric parameters of EK Eri were
determined by adjusting the microturbulence (), ,
and
until no correlation was found between the abundances
of Fe I and their equivalent
width or the excitation
potential and with the additional requirement that the mean Fe
abundance
is the same found from neutral and ionized lines.
The resulting parameters are K,
and
km s-1.
We estimated the uncertainties by perturbing ,
and
in a grid around the derived values by
km s-1,
K,
and dex,
respectively, and measuring the change in the correlations described
above (see Bruntt et al.
2008). These uncertainties are
``internal'' errors since we assume the model atmosphere describe
the actual star. We have added an addition systematic
error of 50 K and 0.1 dex to
and
in Table 1.

The abundance pattern for 17 elements is shown in
Fig. 6
and listed in Table 3.
The uncertainties on the abundances are the quadratic sum of the
standard deviation of the mean value
and the contribution due to the uncertainty on the model parameters (
dex).
Our estimate of the metallicity is based on the mean
abundance of iron-peak elements with at least 10 lines (Si, Ti, V, Cr,
Fe and Ni)
giving
for the combined spectrum.

From Fig. 6
we do not see evidence for peculiarities in the abundances, which might
otherwise have hinted at an earlier phase as a chemically peculiar
star. Any abundance peculiarities present initially in the photosphere
seem to have been efficiently homogenized with the deeper layers
as the star evolved off the main sequence.

Our results on the fundamental parameters
and abundances are different from what we found in Paper I at
a level that may seem incompatible with the uncertainties:
is 100 K cooler and
is 0.1 dex lower in our new analysis.
The quality of the spectrum used in Paper I was lower and
therefore only half as many lines could be used
(
near 6300 Å in Paper I compared to 1300 in our new
spectrum).
Both analyses were done by measuring abundances differentially to the
same lines in a spectrum of the Sun. In Paper I we used a sky
spectrum as a proxy for the Sun, while in the current analysis we use
the high-quality spectrum from Kurucz
et al. (1984). This difference could introduce a
systematic offset.
It can be debated whether it is valid to use
the differential approach since EK Eri is a more massive and
evolved star
compared to the Sun. In fact, the differences we find in
and
may reflect realistic uncertainties to be adopted for the star.

A more intriguing idea is that the lower
we find is due to
a higher amount of spot coverage at the time of observation.
To investigate the evidence for changes in
at different rotational epochs, we have constructed spectra
by combining subsets of our new data. These four spectra, designated E1
to E4,
are listed in Table 4
along with the parameters determined from them. The time intervals
corresponding to these spectra are indicated in Fig. 8. Firstly, we do
not see any significant abundance variation in the E1-E4 spectra.
Secondly,
seems to be systematically lower when the star is faint (which would be
expected if it would be an effect of changes in spot coverage).
However, we do not claim that the slight differences between the
E1...E4 spectra
are real, since the variation of
and
are only 40 K and 0.1 dex respectively,
which are well within the uncertainties of the main analysis.
Although the relative change in
can be measured very precisely,
we must consider
that the model atmospheres have been interpolated from the relatively
coarse grid of Heiter
et al. (2002) which
has a finite grid step size of 200 K in
and 0.2 dex in .
This is the main argument that we do not claim the slight differences
in
to be a physical phenomenon.

Figure 6:

The abundance pattern of EK Eri measured relative to the Sun.
Circles are mean abundances of neutral elements and squares are for
singly ionized species.

Table 4:
Results of the spectroscopic analysis using sub-sets E1-E4 of the HARPS
spectra.

Figure 7:

Synthetic profiles of the Fe I
6151.6 Å line fitted to the CES spectrum.
The upper panel shows the observed spectrum in
grey, and two fits using
the values of
and
listed. The residuals of the fits are plotted underneath. The
lower panel shows the
contours as a function of
and
for the same Fe line with the position of the two plotted fits marked
by filled circles (red and black).
Indicated with dashed vertical lines are the limits on
imposed by the stellar radius and the
and
alternatives.

3.4 Mass estimates

To estimate the mass of the star we followed the approach of Stello et al. (2008)
by applying a scaling relation of ,
which includes the luminosity, effective temperature, and mass.
To estimate the luminosity we use the V magnitude,
bolometric correction and the parallax. The magnitude
is the brightest measured magnitude, hence
we assume it corresponds to no or low spot activity.
This value is probably only accurate to within mag.
We adopted a bolometric correction from the tables in Bessell et al. (1998)
for the
and
obtained from the spectroscopic analysis.
Finally, we use the parallax mas
(van Leeuwen 2007) to
get
and .
To obtain the mass we use the observed value of Hz,
and get .

Another way to estimate the mass is to compare the location of the star
in the HR diagram with theoretical isochrones
or evolutionary tracks. Stello
et al. (2009) compared different methods to estimate
the fundamental parameters of stars from basic photometric indices. We
adopted their SHOTGUN approach, which uses BASTI isochrones
(Pietrinferni et al.
2004) and models without overshoot, taking as input the
metallicity, effective temperature,
and luminosity.
The output from the program is the mass, radius and age of the star:
,
,
and age Gyr.
>From M and R we get ,
which is very similar to the value from
the spectral analysis in Sect. 3.3. Including
overshoot in the BASTI models results in only slightly different
values;
,
,
age Gyr,
and .
In Table 1
we list the results using the standard BASTI isochrones without
overshoot.

These two independent mass estimates are in rough agreement. A
seismologically determined mass estimate is potentially more accurate
than isochrone fitting
if one can measure the large separation
or
the position of maximum power
(Basu et al. 2009).
Since our data do not allow
us to measure
and the error on
is rather large, we adopt the mass estimate based on the SHOTGUN
approach.

4 Discussion

4.1 Rotation period, ,
and radius

The rotational velocity of EK Eri is notoriously hard to
measure due the the very slow rotation. The first accurate
determination
was done by Strassmeier
et al. (1999) from R=120 000
spectra, from which they derived km s-1
with an adopted value of the macroturbulent velocity km s-1,
with a Fourier analysis of the line profile as a check of the validity
of the
adopted .
In Paper I we derived km s-1
by simple profile fitting to 1D atmospheric model spectra,
assuming a similar value of the macroturbulence.

As we will argue in Sect. 4.2, we suggest
that the rotational period of EK Eri
could be twice the photometric period, i.e. close to 617 d.
We have derived
(see Sect. 3.2),
which is in good agreement with the value of
we derive using Eq. (1) from Strassmeier
et al. (1999). Adopting ,
this means that the radius of the star will
be ,
where we assume that .
With the derived radius of ,
this translates to an expectation for the measurement of ,
namely km s-1.
If on the other hand ,
then
must be less than 0.80 km s-1,
as already noted by Strassmeier
et al. (1999), and this value can thus be regarded
as a safe upper limit on .

Using the high-resolution CES data, and our newly derived
stellar parameters, we have attempted to derive
and .
For this analysis we chose 7 lines from the two CES spectra, chosen to
be non-blended based on the line list
obtained from VALD (Kupka
et al. 1999) and visual evaluation of the line
symmetry. The synthetic spectrum is broadened with
different values of
and
in a fine grid in the range 0-4 km s-1
for both parameters, and the
is calculated.
At values of km s-1
the fits start to deviate significantly and we thus note that a
value of km s-1
(originally from Gray 1992,
as a ``typical'' value) is inconsistent with our results.

There is a strong correlation between the broadening caused by
rotation and by macroturbulence which
is very difficult to separate even though their exact forms are
slightly different.
In general we find a continuum of solutions, as illustrated in
Fig. 7,
which shows the reduced
surface for
one spectral line for different values of
and .
The
surfaces for the other lines look very similar.
We find that
is so low that it cannot be determined reliably and only
upper limits can be set. Note that we do not include Zeeman broadening,
which in any case would contribute to an even lower limit on .

4.2 Radial velocity and activity variations

It is well known that stellar magnetic activity affects the shape of
spectral lines and thereby the apparent RV (Gray 2005,1988), which again affects the
ability to detect planets by the Doppler technique.
The best direct measure of activity available in the HARPS spectra are
the emission cores of the calcium H and K lines, and for each spectrum
we have derived the activity index
following the procedure of Paper I.
It was shown by Queloz
et al. (2001)
that the bisector inverse slope (BIS) of the CCF
is a good qualitative measure of the distortion of the spectral lines
caused by activity, and we have calculated the BIS
for all CCFs following the procedure of Dall
et al. (2006). The series of RV, BIS and
are shown
in Fig. 8
along with the corresponding part of the photometric light curve.
In Figs. 9
and 10
we show the correlations of RV with activity index
and BIS.

Figure 8:

Comparison of the photometric and spectroscopic time series over the 3
years of simultaneous monitoring.
The colors and the symbols in the 3 lower plots are: blue ():
2004-09-30-2005-03-20, green (): 2005-08-10-2006-04-02, red ():
2006-07-17-2006-09-08, orange (): 2006-11-06-2007-04-02
(except the three-night run in March 2007),
and grey (,
at 2007.4):
2007-03-03-05 (3 nights of high-cadence data, cf. Fig. 2). The expected
times of photometric minima (vertical dotted lines) and maxima
(vertical dashed lines) are indicated.
The data segments E1-E4 are marked (see Table 4 and
Sect. 3.3).

Activity index
vs. RV using the same symbols as in Fig. 8. The lines and
arrows indicate the evolution with time.
The small grey points inside the box are the 3 nights of high-cadence
data.
The change in activity level during these nights is only about 0.02 in
which is much smaller than the long-term variations.
The clear correlation between activity level and RV
shows that the majority of the RV variations come from activity-related
causes. The overall level seems to have increased between 2004-05 and
2006-07.

BIS vs. RV with the same symbols in Figs. 8 and 9.
The night-to-night variation is very small and comparable with the
single-night scatter. Thus, the larger scale variation in BIS, which
correlates well with RV, is real and attributable to line profile
variations.

From Figs. 8
and 9, it is
clear that the large scale RV variations are very well correlated with
the activity as measured by .
As the star gets dimmer, it gets redder (as noted by Strassmeier et al. 1999)
and the chromospheric activity gets higher, both facts pointing to cool
spots with overlaid plages as the cause of the variation.
The variation of RV seems to be correlated with the photometric
variation as well, although the correspondence is
not always straightforward and the RV amplitude varies differently from
the light amplitude.

The BIS shows in general
a strong correlation with RV as does .
However, there seem to be significant
deviations: the red points (
symbols), corresponding to about two months in late 2006 only, show
a very distinct behavior in BIS, essentially reversing the sign of the
RV-BIS correlation. This behavior is not seen in the direct activity
measure .
The last observing period (
symbols) exhibit variations in
RV and which is
overall anti-correlated with the light variation, although much more
complex. The BIS varies very little during this period, but like
shows a quite complex behavior. This is in sharp contrast to the
behavior of the blue points
(
symbols) where all parameters display large-amplitude, almost
textbook-like behavior with
in anti-correlation with the light variation, and the BIS and RV being
phase shifted with respect to the light. Note that the internal
precision on individual data points is very high
as evidenced by the low scatter of the three-night run.

It is also worth noting that the obvious strong variation of
bisector shape is in apparent contradiction to the finding of
Saar & Donahue (1997)
who found that the BIS loses its diagnostic power for extremely low .
Their model results were confirmed
recently by Desort et al.
(2007), who found that a
less than the spectrograph resolution (which is certainly the case for
EK Eri)
should yield a constant bisector shape. This was confirmed by Bonfils et al. (2007)
and Huélamo et al. (2008)
who found
negligible BIS-RV correlations for the slow (
km s-1)
rotators EK Eri and EK Eri, respectively.
This is in contrast to our findings as evident from Fig. 10. Presently, we
can offer no explanation of this.

Two other aspects are worth pointing out. First, while Desort et al. (2007)
find the BIS-RV slope to be negative in all their simulations, we are
clearly seeing mostly positive slopes, although with at least one
period of negative slope as well. These authors however presented only
a few models and only for main sequence stars.
Secondly, while Bonfils
et al. (2007) find a clear loop pattern in the plot
of RV versus Ca II H+K emission, which they
interpret
in terms of rotational modulation, our results are less clear, mostly
due to the incomplete phase coverage.
However, from the time evolution of the -RV and
BIS-RV relations as depicted in Figs. 9 and 10
we can see that there appears to be different segments of similar loop
patterns for different observing seasons, which could
mean that these segments of the loops are not associated with the same
spot regions. Indeed, if the star is viewed equator-on, then one spot
or spot group is not enough to produce a sinusoidal light curve, much
less the radial velocity and activity variations.

One possible conceptual model of EK Eri is that of an
oblique rotator seen close to ,
with the magnetic axis of a dipolar field tilted with respect to the
rotation axis. This assumes that we approximate the star as a single
dipole,
which we believe is a reasonable approximation, taking into account
also
the results of Aurière
et al. (2008). Assuming that the magnetic poles are
associated with the spots,
we propose a model of EK Eri involving two large spots or
spot-covered areas, located 180
opposite each other.
Although Aurière et al.
(2008) observed no sign changes in the average longitudinal
field over the
photometric phase ,
which would correspond to the hypothetical two spots gradually
rotating into (respectively, out of) view, we note that their
observations were performed during a period
where EK Eri seemed to be undergoing another cycle change similar to
the one of 1987-1992. During these periods
the light amplitude is low, indicating that spots are small and
possibly dominated by dynamo-related activity
rather than by the large scale dipole field.

In this scenario, the light curve minima correspond to a spot
facing the observer, which happens twice per rotation. Hence, the
rotation period
of the star is not equal to ,
but rather twice that. This would mean that the true latitude-averaged
rotation period of EK Eri is d.
The possibility of
being two or even three times the photometric period was also mentioned
by Aurière et al. (2008)
but
not investigated further. Such a relationship has been observed for the
Sun (e.g., Durrant
& Schroeter 1983) where the rotation of active
regions
across the solar disk gives rise to a 13-day period, i.e., half the
solar rotation period. The phenomenon of
has also been suggested for Procyon (Arentoft
et al. 2008).
Note that the period
would not show up in the period analysis unless the spots were of
uneven size and stable over several rotations, which is clearly not the
case.

Although simple, this would explain the differences seen in
the -RV loop
patterns which seem to change direction
for every second loop, while also qualitatively explaining the
appearance of the light curve. Of course, we are assuming that the
field is poloidal, that the spots are associated with the large scale
field, and that the star is seen equator-on. We have however not been
able to produce a spot model that could reproduce the light curve
satisfactorily
using this simple geometry, and the true structure of the magnetic
field and the activity of EK Eri is likely far more
complex.

4.3
Photometric period changes

As evident from the light curve (Fig. 1) and as noted
by Strassmeier
et al. (1999), the photometric period
of EK Eri is not stable. In fact, based on the latest data,
the star appears to be going through a phase similar to the period
1987-1992 where the light variations almost disappeared and the period
changed significantly. While this may be explained by distinct magnetic
cycles, other explanations have been proposed in view of the unusually
long period.
One possibility proposed hypothetically by Strassmeier et al. (1990)
was that the star is seen pole-on and that the long
period reflects activity cycles on a rapidly rotating star.
Alternatively, they suggested a strong internal
rotation gradient to explain the activity in terms of a classical
dynamo. Stepien (1993)
proposed the elegant solution that EK Eri
is the descendant of a magnetic Ap star. Up to 10% of Ap stars have
rotation periods longer than 100 d (e.g., Mathys 2008), and a significant
fraction of the field should be able to survive main sequence evolution
(Moss 2003).

4.4
Magnetic field geometry, and low-amplitude stellar oscillations

Aurière et al. (2008)
observed modulations of the magnetic field strength with rotational
phase, but they
were unable to relate it to the photometric period.
In particular, they observed a peak in the field strength at one phase,
which they
assign as their phase zero point. From our ephemeris given in
Table 1
we find that their
zero phase corresponds to
in our data i.e., very close to quarter photometric phase.

In our RV monitoring, we have covered the interval from time
of photometric minimum (
)
to first quarter
phase on two segments of the light curve; in 2004-2005 and 2006-2007
(see Fig. 8).
In the first segment, all parameters vary smoothly. As we noted, this
is the classical textbook behavior. In the second segment, the activity
level as inferred by
is overall higher, with the maximum level likely to have been reached
just before 2006-11-06, but after the time of light minimum, i.e.
around -0.2.
It is likely then, that the high field strength measured by Aurière et al. (2008)
corresponds to a similar
high value of
and vice versa. Furthermore, around
in 2006-2007, the BIS changes direction
for a short while,
which may indicate an additional contribution on top of the main spot
(group), possibly a smaller bright spot, or
the emergence of a short lived spot. If an
dynamo can work locally to augment a large scale seed field
at small scales on the stellar disk (a mechanism proposed for AGB stars
by Soker & Zoabi 2002),
then this spike could be caused by spots with completely different
characteristics in terms of size and temperature, appearing alongside
the semi-static fossil field-induced spots responsible for the main
light modulation.

The magnetic field geometry suggested by Aurière et al. (2008)
is close to a dipole, which makes intuitive sense for
a remnant static field. One may expect a large scale magnetic field to
have a stabilizing effect on the overall shape
of the star, i.e., resisting deformations caused by the oscillations.
As noted by Braithwaite (2009),
a dominantly poloidal
field tends to align the magnetic axis perpendicular to the rotation
axis, thereby contributing to making the star
oblate, even for very slow rotation.
With this in mind, the surprisingly low oscillation
amplitude-per-radial-mode (cf. Sect. 3.2.1)
may have a logical explanation. In most stars, the l=1
modes are expected to have the highest
amplitudes, but if the low-degree modes are suppressed, then the bulk
of the oscillations will be
higher degree modes. Assuming an expected amplitude per mode of 0.5 m s-1
(Fig. 8 of Kjeldsen
et al. 2008)
we find a spatial response scaling factor Sl/S0
= 0.3. Comparing with Table 1 of Kjeldsen
et al. (2008) we then
propose that the bulk of the oscillations in EK Eri could be
higher degree modes of l=3, 4. Coincidentially, for
the rapidly oscillating magnetic Ap stars (roAp), which are possible
progenitors of EK Eri,
there is emerging evidence for interaction between the magnetic field
and the oscillation driving mechanism
(e.g., Saio et al. 2010,
and references therein).

Of course our time coverage is very poor and the discrepancy
may very well be due to mode beating and the stochastic
nature of mode excitation. Obviously, longer observing runs are
required to test this scenario. It is interesting to note that the
magnetic field measurement of Aurière
et al. (2008) as well as our
asteroseismic results were obtained while the star was apparently
entering another ``bright'' phase akin to the 1987-1992
period. In order to test the interaction between the magnetic field and
the oscillations it would be highly
desirable to conduct asteroseismic measurements in periods of high
field strength as well as in periods of relatively
low field strength.

5 Conclusions

We have presented results from an intensive monitoring of the active
sub-giant star EK Eri.
We have used photometric data covering 30 years and spectroscopic data
probing long-term variation in activity during 3 years and high-cadence
radial-velocity monitoring from 3 nights. >From the photometry
we have refined the rotation period and the ephemeris as listed in
Table 1.
Also, from 3 half-nights of high-cadence RV monitoring we have detected
solar-like oscillations
in a late-type spotted sub-giant star for the first time. While
oscillations have been detected in a number of late-type giants
(e.g., Hatzes
& Zechmeister 2008; Hekker et al. 2009)
and for mildly active solar-type stars (e.g., Pollux: Aurière
et al. 2009; Hatzes & Zechmeister
2007), this is the first detection for a sub-giant hosting a
strong magnetic field.
Unfortunately our data do not allow us to resolve individual
frequencies.
We measure an amplitude per radial mode of 0.15 m s-1
at a position of maximum power Hz.
This amplitude is at least a factor of 3 lower than expected and, if
confirmed, may mean that
the magnetic field has a strong
stabilizing effect on the stellar geometry, essentially favoring
high-degree oscillation modes in the presence of
a near-dipole magnetic field. In that case, the interpretation of the
asteroseismic data may become more difficult for this and other
sub-giant stars with similar magnetic properties. A longer time series
is required in order to obtain quantitative results.
For reference purposes, having accurate oscillation data for a
non-active star
at the position in the HR diagram of EK Eri would be highly
desirable. Unfortunately, neither
CoRoT nor Kepler apparently covers this.

Based on the roughly sinusoidal shape of the light curve, the
likely very high inclination, the field geometry suggested by Aurière et al. (2008),
and the behavior of the activity indicators as function of RV, we
suggest a conceptual model of EK Eri with two large
low-latitude spot covered areas approximately
apart on a star viewed equator-on. In this scenario, the rotational
period is twice the photometric period, thus d.
We note however, that a simple two-spot model is not able to account
for all the seasonal light variations observed, mostly due to the
unknown spot lifetimes, sizes and longitudes.

Regardless of the rotation period, the measured values of
both from this work and from the literature are
inconsistent with the derived radius of the star. Both the radius
derived from asteroseismology and from the spectral analysis set strict
upper limits on
which are lower than previous estimates.
We thus conclude that the
is too low to be reliably measured with available spectrographs.

Based on high-quality HARPS spectra we have derived the
atmospheric parameters of EK Eri to very high
precision. The abundance pattern for 17 analysed elements is very
similar to the Sun,
and we detect no anomalies that could otherwise be attributed
to an earlier evolutionary state as a magnetic Ap star.
However, in order to argue
for or against EK Eri being a descendant of a magnetic Ap
star, stronger
constraints on the mass and evolutionary state are needed. Further
seismic studies,
preferably at varying rotational phases, may deliver such constraints
in terms of accurate asteroseismic mass and radius measurements,
and are also needed to probe the possible link between solar-like
oscillations and the magnetic field.

Acknowledgements

D.S. would like to acknowledge support from the
Australian Research Council.
T.D. acknowledges support by the Gemini Observatory, which is operated
by the Association of Universities for Research in Astronomy, Inc., on
behalf of the international Gemini partnership of Argentina, Australia,
Brazil, Canada, Chile, the United Kingdom, and the United States of
America.
This research has made use of the SIMBAD database, operated at CDS,
Strasbourg, France.
We would like to thank the anonymous referee for comments and
suggestions that have improved the paper.

Footnotes

... EK Eridani

Based on data from the HARPS spectrograph at the La Silla
Observatory, European Southern Observatory, obtained under programs IDs
77.C-0080 and 78.C-0233, and on data from the CES spectrograph obtained
from the ESO Science Archive Facility.

All Tables

Table 4:
Results of the spectroscopic analysis using sub-sets E1-E4 of the HARPS
spectra.

All Figures

Figure 1:

The photometric time series in V from 1979-2009 and
in V-I from 1996-2009. The
expected times of V maxima are indicated with
dashed lines. The inset shows the power spectrum from the period
analysis (see Sect. 3.1).

RV vs. BJD for the three nights of high-cadence spectra used for the
asteroseismic analysis. The mean error bar is shown at the
lower left corner.
The three-night run displays clear variation, but with period
variations from
night-to-night, likely due to mode beating.

A section of the combined HARPS spectrum illustrating the high S/N.
The observed and synthetic spectra are shown with solid and dot-dashed
lines, respectively.
The main lines contributing to the spectrum are marked by vertical
lines. The solid vertical lines mark the least blended lines that are
used in the abundance analysis.

Synthetic profiles of the Fe I
6151.6 Å line fitted to the CES spectrum.
The upper panel shows the observed spectrum in
grey, and two fits using
the values of
and
listed. The residuals of the fits are plotted underneath. The
lower panel shows the
contours as a function of
and
for the same Fe line with the position of the two plotted fits marked
by filled circles (red and black).
Indicated with dashed vertical lines are the limits on
imposed by the stellar radius and the
and
alternatives.

Comparison of the photometric and spectroscopic time series over the 3
years of simultaneous monitoring.
The colors and the symbols in the 3 lower plots are: blue ():
2004-09-30-2005-03-20, green (): 2005-08-10-2006-04-02, red ():
2006-07-17-2006-09-08, orange (): 2006-11-06-2007-04-02
(except the three-night run in March 2007),
and grey (,
at 2007.4):
2007-03-03-05 (3 nights of high-cadence data, cf. Fig. 2). The expected
times of photometric minima (vertical dotted lines) and maxima
(vertical dashed lines) are indicated.
The data segments E1-E4 are marked (see Table 4 and
Sect. 3.3).

Activity index
vs. RV using the same symbols as in Fig. 8. The lines and
arrows indicate the evolution with time.
The small grey points inside the box are the 3 nights of high-cadence
data.
The change in activity level during these nights is only about 0.02 in
which is much smaller than the long-term variations.
The clear correlation between activity level and RV
shows that the majority of the RV variations come from activity-related
causes. The overall level seems to have increased between 2004-05 and
2006-07.

BIS vs. RV with the same symbols in Figs. 8 and 9.
The night-to-night variation is very small and comparable with the
single-night scatter. Thus, the larger scale variation in BIS, which
correlates well with RV, is real and attributable to line profile
variations.

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